A HistoryMathematicsCarl

R Boyer

WILEY INTERNATIONAL EDITION

MATHEMATICAL CENTERS

IN

THE THALASSIC AGE

1.

Syracuse

AbderaAlexandria

10

(Democritus)(Euclid, Heron, Ptolemy,

2.

CrotonaElea

25811

Pappus

Menelaus and others)3.

4.5.

RomeTarentum CyreneElis

Athens Byzantium ChalcedonChalcis

(Plato, Theaetetus)

(Proclus)

12

(Xenocrates)(Iamblichus)

6.7.

2316

8. 9.

AthensStagira

10. 11. 12. 13. 14. 15.

Abdera Byzantium ChalcedonNicaea Cyzicus

Chios Cnidus Crotona CyreneCyzicusEleaElis

(Hippocrates)

202 6143

(Eudoxus)(Pythagoras)

(Theodorus, Eratosthenes)(Callippus)

(Parmenides, Zeno)(Hippias)

7

GerasaMiletus

2419 13

(Nicomachus)(Thales)

PergamumChios

16.17. 18. 19. 20.

Samos SmyrnaMiletus

Nicaea Perga

(Hipparchus)(Apollonius) (Apollonius)

2215 21

PergamumRhodes

(Eudemus, Geminus)(Boethius)

Cnidus

RomeSamos SmyrnaStagira

41718

21.22.

RhodesPerga

(Pythagoras, Conon, Aristarchus)

(Theon)(Aristotle)

23. Chalcis

9

Gerasa 25. Alexandria 26. Syene24.

Syene Syracuse

261

(Eratosthenes)

(Archimedes)(Pythagoras, Archytas, Philolaus[?])

Tarentum

5

A

History of Mathematics

Carl B. Boyer

Professor of Mathematics Brooklyn College

A Historyof

Mathematics

JOHN WILEY& SONS, INC.

New York London Sydney

|

ACCESSiqdk'o.

CLAJ

Aj nr

"

f'

29 MAR 1979 "7

N

/

10Copyright

1968 by John Wiley

&

Sons, Inc.

All rights reserved.

No part of this book may

be

reproduced by any means, nor transmitted, nor translated into a machine language without thewritten permission of the publisher.

Library of Congress Catalog Card

Number:

68-1 650i

Printed in the United States of America

To

the

MemoryHowardand

of

My Parents

Franklin Boyer

Rebecca Catherine (Eisenhart) Boyer

PrefaceNumerous histories of mathematics have appeared during this century, many of them in the English language. Some are very recent, such as J. F. Scott's A History of Mathematics 1 a new entry in the field therefore should;

have characteristics not already present in the available books. Actually, few of the histories at hand are textbooks, at least not in the American sense of the word, and Scott's History is not one of them. It appeared, therefore, that there was room for a new book one that would meet more satis-

preferences and possibly those of others. The two-volume History of Mathematics by David Eugene Smith 2 was indeed written "for the purpose of supplying teachers and students with afactorily

my own

usable textbook on the history of elementary mathematics," but it covers too wide an area on too low a mathematical level for most modern collegecourses,

and

it

is

lacking in problems of varied types. Florian Cajori's2,

still is a very helpful reference work; but it is not adapted to classroom use, nor is E. T. Bell's admirable The Development of Mathematics. 4 The most successful and appropriate textbook today appears to be Howard Eves, An Introduction to the History of Mathematics, 5 which I have used with considerable satisfaction in at least a dozen classes since it

History of Mathematics

appeared in 1953. I have occasionally departed from the arrangement of topics in the book in striving toward a heightened sense of historicalfirst

mindedness and have supplemented the material by further reference to the contributions of the eighteenth and nineteenth centuries especially by the use of D. J. Struik, A Concise History of Mathematics. 6 The reader of this book, whether a layman, a student, or a teacher of a course in the history of mathematics, will find that the level of mathematical background that is presupposed is approximately that of a college junior or senior, but the material can be perused profitably also by readers with either stronger or weaker mathematical preparation. Each chapter ends with a set of exercises that are graded roughly into three categories. Essay questions that are intended to indicate the reader's ability to organize and put into his own words the material discussed in the chapter are listed first. Then followrelatively easy exercises that require the proofs of

mentioned1

in the

some of the theorems chapter or their application to varied situations. Finally,

23

London: Taylor and Francis, 1958. Boston: Ginn and Company, 1923-1925. New York: Macmillan, 1931, 2nd edition.

45

6

New New New

York: McGraw-Hill, 1945, 2nd edition. York: Holt, Rinehart and Winston, 1964, revised York: Dover Publications, 1967, 3rd edition.

edition.

viii

PREFACE

which are either more difficult or require methods that may not be familiar to all students or all readers. The exercises do not in any way form part of the general exposition and can be disregarded by the reader without loss of continuity. Here and there in the text are references to footnotes, generally bibliographical, and following each chapter there is a list of suggested readings. Included are some references to the vast periodical literature in the field,there are a few starred exercises,specializedforit is

of material available in good libraries. Smaller college librariesable to provideall

not too early for students at this level to be introduced to the wealth may not beof these sources, but

it is well for a student to be aware of beyond the confines of his own campus. There are references also to works in foreign languages, despite the fact that some students, hopefully not many, may be unable to read any of these.

the larger realms of scholarship

who have a reading knowledge of a foreign language, the inclusion of references in other lanBesides providing important additional sources for those

guageslike,

mayin,

help to break

down

the linguistic provincialism which, ostrich-

takes refuge in the mistaken impression that everything worthwhile

appeared

or has been translated into, the English language.differs

The present work

from the most successful presently available

textbook in a stricter adherence to the chronological arrangement and a stronger emphasis on historical elements. There is always the temptation in a class in history of mathematics to assume that the fundamental purpose of the course is to teach mathematics. departure from mathematical standards is then a mortal sin, whereas an error in history is venial. I have striven to avoid such an attitude, and the purpose of the book is to present the history of mathematics with fidelity, not only to mathematical structure and exactitude, but also to historical perspective and detail. It would be folly, in a book of this scope, to expect that every date, as well as every decimal point, is correct. It is hoped, however, that such inadvertencies as may survive beyond the stage of page proof will not do violence to the sense of history, broadly

A

understood, or to a sound view of mathematical concepts.strongly emphasized that this single

It

cannot be too

volume in no way purports to present the history of mathematics in its entirety. Such an enterprise would call for the concerted effort of a team, similar to that which produced the fourth volume of Cantor's Vorlesungen uber Geschichte der Mathematik in 1908 and brought the story down to 1799. In a work of modest scope the author must exercise judgment in the selection of the materials to be included, reluctantly restraining the temptation to cite the work of every productive mathematician it will be an exceptional reader who will not note here what;

he regards as unconscionable omissions. In particular, the last chapter attempts merely to point out a few of the salient characteristics of thetwentieth century. In the field of the history of mathematics perhaps nothing

PREFACEis

ix

more

who would complete

to be desired than that there should appear a latter-day Felix Klein for our century the type of project Klein essayed for to

for the nineteenth century, but did not live to finish.

A

published work

is

some

extent like an iceberg, for

what

is

visible

appears until the author has lavished time on it unstintingly and unless he has received encouragement and support from others too numerous to be named individually. Indebtedness in my case begins with the many eager students to whom I have taught the history of mathematics, primarily at Brooklyn College, but also at Yeshiva University, the University of Michigan, the University ofCalifornia (Berkeley), and the University of Kansas.

constitutes only a small fraction of the whole.

No book

At the University of

Michigan, chiefly through the encouragement of Professor Phillip S. Jones, and at Brooklyn College through the assistance of Dean Walter H. Mais and Professors Samuel Borofsky and James Singer, I have on occasion enjoyed a reduction in teaching load in order to work on the manuscript of this book. Friends and colleagues in the field of the history of mathematics, including Professor Dirk J. Struik of the Massachusetts Institute of Technology, Professor Kenneth O. May at the University of Toronto, Professor Howard Eves of the University of Maine, and Professor Morris Kline at New York University, have made many helpful suggestions in the preparation of the book, and these have been greatly appreciated. Materials in the

books and articles of others have been expropriated freely, with little acknowledgment beyond a cold bibliographical reference, and I take thisopportunity to express to these authors my warmest gratitude. Libraries and publishers have been very helpful in providing information and illustrait has been a pleasure to have worked of John Wiley and Sons. The typing of the final copy, as well as of much of the difficult preliminary manuscript, was done cheerfully and with painstaking care by Mrs. Hazel Stanley of Lawrence, Kansas. Finally,

tions needed in the text; in particular

with the

staff

I

must express deep gratitude to a very understanding wife, Dr. Marjorie N. Boyer, for her patience in tolerating disruptions occasioned by the development of yet another book within the family.Brooklyn,

New

York

Carl

B.

Boyer

January 1968

ContentsChapter1

I.

Primitive Originsbases. 3

1

The concept of number. 2 Early numberorigin of counting.

Number language and

the

4 Origin of geometry.

Chapter1

II.

Egypt

g

Early records. 2 Hieroglyphic notation. 3 Ahmes papyrus. 4 Unit fractions. 5 Arithmetic operations. 6 Algebraic problems. 7 Geometrical problems.

8

A

trigonometric ratio. 9

Moscow

papyrus. 10 Mathematical weaknesses.

Chapter1

III.

Mesopotamia

26

Cuneiform records. 2 Positional numeration. 3 Sexagesimal fractions. 4 Fundamental operations. 5 Algebraic problems. 6 Quadratic equations. 7 Cubic equations. 8 Pythagorean triads. 9 Polygonal areas. 10 Geometry as applied arithmetic. 11 Mathematical weaknesses.

Chapter1

IV.

Ionia

and the Pythagoreansof Miletus. 3 Pythagoras of Samos. 4 The Pythagorean mysticism. 6 Arithmetic and cosmology. 7 Figurate

48

Greek

origins. 2 Thales

pentagram. 5

Number

numbers. 8 Proportions. 9 Attic numeration. 10 Ionian numeration. 11Arithmetic andlogistic.

Chapter V.1

The Heroic Age

69

Centers of activity. 2 Anaxagoras of Clazomenae. 3 Three famous problems.

4 Quadrature of lunes. 5 Continued proportions. 6 Hippias of Ellis. 7 Philolaus and Archytas of Tarentum. 8 Duplication of the cube. 9 Incommensurability. 10 The golden section. 11 Paradoxes ofZeno. 12 Deductive reasoning.

13 Geometrical algebra. 14 Democritus of Abdera.

Chapter1

VI.

The Age of Plato and Aristotleliberal arts.

91

The seven

2 Socrates. 3 Platonic solids. 4 Theodorus of Cyrene. 5 Platonic arithmetic and geometry. 6 Origin of analysis. 7 Eudoxus of Cnidus. 8 Method of exhaustion. 9 Mathematical astronomy. 10 Menaechmus. 11 Duplication of the cube. 12 Dinostratus and the squaring of the 13 Autolycus of Pitane. 14 Aristotle. 15 End of the Hellenic period.

circle.

Chapter1

VII.

Euclid of Alexandria

111

Author of the Elements. 2 Other works. 3 Purpose of the Elements. 4 Definitions and postulates. 5 Scope of Book I. 6 Geometrical algebra. 7 Books III and IV. 8 Theory of proportion. 9 Theory of numbers. 10 Prime and perfect numbers. 11 Incommensurability. 12 Solid geometry. 13 Apocrypha. 14Influence of the Elements.

xii

CONTENTSVIII.

Chapter1

Archimedes of SyracuseLawof the lever. 3

134The

The

siege of Syracuse. 2

The

hydrostatic principle. 4

Sand- Reckoner. 5 Measurement of the circle. 6 Angle trisection. 7 Area of a parabolic segment. 8 Volume of a paraboloidal segment. 9 Segment of a sphere. 10 On the Sphere and Cylinder. 11 Book of Lemmas. 12 Semiregularsolids and trigonometry. 13 The

Method. 14 Volume of a sphere. 15 Recovery

of the Method.

Chapter1

IX.

Apollonius of Perga

157

Lost works. 2 Restorations of lost works. 3 The problem of Apollonius. 4 Cycles and epicycles. 5 The Conies. 6 Names of the conic sections. 7 The double-napped cone. 8 Fundamental properties. 9 Conjugate diameters.10 Tangents and harmonic division. 11 The three-and-four-line locus. 12 Intersecting conies. 13 Maxima and minima, tangents and normals.

14 Similar conies. 15 Foci of conies. 16 Use of coordinates.

Chapter1

X.

Greek Trigonometry and Mensuration

176

Early trigonometry. 2 Aristarchus of Samos. 3 Eratosthenes of Cyrene. 4 Hipparchus of Nicaea. 5 Menelaus of Alexandria. 6 Ptolemy's Almagest. 7 The 360 degreecircle. 8 Construction of tables. 9 Ptolemaic astronomy. 10 Other works by Ptolemy. 11 Optics and astrology. 12 Heron of Alexandria. 13 Principle of least distance. 14 Decline of Greek mathematics.

Chapter XI.1

Revival and Decline of Greek Mathematics

196

Applied mathematics. 2 Diophantus of Alexandria. 3 Nicomachus of Gerasa. 4 The Arithmetica of Diophantus. 5 Diophantine problems. 6 The place of Diophantus in algebra. 7 Pappus of Alexandria. 8The Collection. 9 Theorems of Pappus. 10 The Pappus problem. 11 The Treasury of Analysis. 12 The Pappus-Guldin theorems. 13 Proclus of Alexandria. 14 Boethius. 15 End ofthe Alexandrian period. 16 cians of the sixth century.

The Greek Anthology. 17 Byzantine mathemati-

Chapter1

XII.

China and India

217

The oldest documents. 2 The Nine Chapters. 3 Magic squares. 4 Rod numerals. 5 The abacus and decimal fractions. 6 Values of pi. 7 Algebra and Horner's method. 8 Thirteenth-century mathematicians. 9 The arithmetic triangle.10 Early mathematics in India. 11 The Sulvasutras. 12 The Siddhantas. 13 Aryabhata. 14 Hindu numerals. 15 The symbol for zero. 16 Hindutrigonometry. 17 Hindu multiplication. 18 Long division. 19 Brahmagupta. 20 Brahmagupta's formula. 21 Indeterminate equations. 22 Bhaskara. 23 The Lilavati. 24 Ramanujan.

Chapter1

XIII.

The Arabic Hegemony

249

Arabic conquests. 2 The House of Wisdom. 3 Al-jabr. 4 Quadratic equations. 5 The father of algebra. 6 Geometric foundation. 7 Algebraic problems. 8 A problem from Heron. 9 Abd al-Hamid ibn-Turk. 10 Thabit ibn-Qurra.11

Arabic numerals. 12 Arabic trigonometry. 13 AbuT-Wefa and alKarkhi. 14 Al-Biruni and Alhazen. 15 Omar Khayyam. 16 The parallel

postulate. 17 Nasir Eddin. 18 Al-Kashi.

CONTENTS

xiii

Chapter XIV.1

Europe

in

the Middle Agestranslation.

272

From Asia

to Europe. 2 Byzantine mathematics. 3

and Gerbert. 5 The century of

The Dark Ages. 4 Alcuin 6 The spread of Hindu-Arabic

numerals. 7 The Liber abaci. 8 The Fibonacci sequence. 9 A solution of a cubic equation. 10 Theory of numbers and geometry. 11 Jordanus Nemor-

12 Campanus of Novara. 13 Learning in the thirteenth century. 14 Medieval kinematics. 15 Thomas Bradwardine. 16 Nicole Oresme. 17 The latitude of forms. 18 Infinite series. 19 Decline of medieval learning.arius.

Chapter XV.1

The Renaissance

297

Humanism. 2 Nicholas of Cusa. 3 Regiomontanus. 4 Application of algebra to geometry. 5 A transitional figure. 6 Nicolas Chuquet's Triparty. 7 Luca Pacioli's Summa. 8 Leonardo da Vinci. 9 Germanic algebras. 10 Cardan'sArs magna. 11 Solution of the cubic equation. 12 Ferrari's solution of the quartic equation. 13 Irreducible cubics and complex numbers. 14 RobertRecorde. 15 Nicholas Copernicus. 16 Georg Joachim Rheticus. 17 Pierre de la Ramee. 18 Bombelli's Algebra. 19 Johannes Werner. 20 Theory of perspective. 21 Cartography.

Chapter XVI.1

Prelude to Modern Mathematics

333

Francois Viete. 2 Concept of a parameter. 3 The analytic

art. 4 Relations between roots and coefficients. 5 Thomas Harriot and William Oughtred. 6 Horner's method again. 7 Trigonometry and prosthaphaeresis. 8 Trigonometric solution of equations. 9 John Napier. 10 Invention of logarithms.

11

Henry

Briggs. 12 Jobst Biirgi. 13 Applied mathematics

and decimal

fractions. 14 Algebraic notations. 15 Galileo Galilei.

16 Values of pi.

17 Reconstruction of Apollonius' On Tangencies. 18 Infinitesimal analysis. 19 Johannes Kepler. 20 Galileo's Two New Sciences. 21 Galileo and theinfinite.

22 Bonaver.tura Cavalieri. 23 The spiral and the parabola.

Chapter XVII.*1

The Time of Fermat and Descartes

367

Leading mathematicians of the time. 2 The Discours de la methode. 3 Invention of analytic geometry. 4 Arithmetization of geometry. 5 Geometrical algebra. 6 Classification of curves. 7 Rectification of curves. 8 Identification of conies. 9 Normals and tangents. 10 Descartes' geometrical concepts. 11 Fermat's loci. 12 Higher-dimensional analytic geometry. 13 Fermat's differentiations. 14 Fermat's integrations. 15 Gregory of St. Vincent. 16 Theory of numbers.17 Theorems of Fermat. 18 Gilles Persone de Roberval. 19 Evangelista

20 New curves. 21 Girard Desargues. 22 Projective geometry. 23 Blaise Pascal. 24 Probability. 25 The cycloid.Torricelli.

Chapter1

XVIII.

A Transitional

Period

404

Philippe de Lahire. 2

Georg Mohr. 3 Pietro Mengoli. 4 Frans van Schooten. 5 Jan de Witt. 6 Johann Hudde. 7 Rene Francois de Sluse. 8 The pendulum clock. 9 Involutes and evolutes. 10 John Wallis. 11 On Conic Sections. 12 Arithmetica infinitorum. 13 Christopher Wren. 14 Wallis' formulas. 15 James Gregory. 16 Gregory's series. 17 Nicolaus Mercator and William Brouncker. 18 Barrow's method of tangents.

xiv

CONTENTS

Chapter XIX.1

Newton and

Leibniz

429

Newton's early work. 2 The binomial theorem. 3 Infinite series. 4 The Method of Fluxions. 5 The Principia. 6 Leibniz and the harmonic triangle. 7 The differential triangle and infinite series. 8 The differential calculus. 9 Determinants, notations, and imaginary numbers. 10 The algebra of logic. 1 1 The inverse square law. 12 Theorems on conies. 13 Optics and curves. 14 Polar and other coordinates. 15 Newton's method and Newton's parallelogram. 16 The Arithmetica universalis. 17 Later years.

Chapter XX.1

The Bernoulli EraThe logarithmic spiral. 3 Probability andinfinite series.

455

The

Bernoulli family. 2

4 L'Hospital's rule. 5 Exponential calculus. 6 Logarithms of negative numbers. 7 Petersburg paradox. 8 Abraham de Moivre. 9 De Moivre's theorem. 10 Roger Cotes. 11 James Stirling. 12 Colin Maclaurin. 13 Taylor's series. 14 The Analyst controversy. 15 Cramer's rule. 16 Tschirnhaus transformations. 17 Solid analytic geometry. 18 Michel Rolle and Pierre Varignon. 19 Mathematics in Italy. 20 The parallel postulate. 21 Divergent

Chapter XXI.1

The Age of EulerLogarithms of negative numbers. 3 Foundation of analysis. series. 5 Convergent and divergent series. 6 Life of d'Alembert.identities.

481

Life of Euler. 2

47

Infinite

The Euler

8 D'Alembert and

limits.

9 Differential equations.

10 The Clairauts. 11 The Riccatis. 12 Probability. 13 Theory of numbers. 14 Textbooks. 15 Synthetic geometry. 16 Solid analytic geometry. 17

Lambert and the

parallel postulate. 18

Bezout and elimination.

Chapter XXII.1

Mathematicians of the French Revolution

510

The age of revolutions. 2 Leading mathematicians. 3 Publications before 1789. 4 Lagrange and determinants. 5 Committee on Weights and Measures. 6 Condorcet on education. 7 Monge as administrator and teacher. 8 Descriptive geometry and analytic geometry. 9 Textbooks. 10 Lacroix on analytic geometry. 11 The Organizer of Victory. 12 Metaphysics of the calculus and geometry. 13 Geometrie de position. 14 Transversals. 15 Legendre's Geometry. 16 Elliptic integrals. 17 Theory of numbers. 18Theory of functions. 19 Calculus of variations. 20 Lagrange multipliers. 21 Laplace and probability. 22 Celestial mechanics and operators. 23Political changes.

Chapter1

XXIII.

The Time of Gauss and Cauchy

544

Early discoveries by Gauss. 2 Graphical representation of complex numbers. 3 The fundamental theorem of algebra. 4 The algebra of congruences. 5Reciprocity and frequency of primes. 6 Constructible regular polygons.

7 Astronomy and least squares. 8 Elliptic functions. 9 Abel's

life

and work.

10 Theory of determinants. 11 Jacobians. 12 Mathematical journals. 13 Complex variables. 14 Foundations of the calculus. 15 Bernhard Bolzano. 16 Tests for convergence. 17 Geometry. 18 Applied mathematics.

CONTENTS

xv

Chapter XXIV.1

The Heroic Age

in

Geometry

572

Theorems of Brianchon and Feuerbach. 2 Inversive geometry. 3 Poncelet's4 Pliicker's abridged notation. 5 Homogeneous coordinates. 6 Line coordinates and duality. 7 Revival of British mathematics. 8 Cayley's ^-dimensional geometry. 9 Geometry in Germany.Bolyais. 13

projective geometry.

10 Lobachevsky and Ostrogradsky. 11 Non-Euclidean geometry. 12 The Riemannian geometry. 14 Spaces of higher dimension. 15

Klein's Erlanger

Programm. 16

Klein's hyperbolic model.

Chapter XXV.1

The Arithmetization of Analysis2 Analytic

598

Fourier

number theory. 3 Transcendental numbers. 4 Uneasiness in analysis. 5 The Bolzano-Weierstrass theorem. 6 Definition of real number. 7 Weierstrassian analysis. 8 The Dedekind "cut". 9 The limit concept. 10 Gudermann's influence. 11 Cantor's early life. 12 The "power"series.

of

infinite sets.

13 Properties of infinite

sets.

14 Transfinite arithmetic.

15 Kronecker's criticism of Cantor's work.

Chapter XXVI.1

The Rise of Abstract Algebrain

620

The Golden Age

mathematics. 2 Mathematics at Cambridge. 3 Peacock, the "Euclid of algebra." 4 Hamilton's quaternions. 5 Grassmann and Gibbs. 6 Cayley's matrices. 7 Sylvester's algebra. 8 Invariants of quadratic forms.

9 Boole's analysis ofPeirces. 12

logic.life

10 Boolean algebra. 11

De Morgan and

the

The

tragic

of Galois. 13 Galois theory. 14 Field theory. 15

Frege's definition of cardinal number. 16 Peano's axioms.

Chapter XXVII.1

Aspects of the Twentieth Century

649

The nature of mathematics. 2 Poincare's theory of functions. 3 Applied mathematics and topology. 4 Hilbert's problems. 5 Godel's theorem. 6Transcendental numbers. 7 Foundations of geometry. 8 Abstract spaces. 9 The foundations of mathematics. 10 Intuitionism, formalism, andlogicism. 11

Measure andin

integration. 12 Point set topology. 13 Increasing

abstraction

15 High-speed computers. 16 Mathematical structure. 17 Bourbaki and the "New Mathematics."14 Probability.

algebra.

General Bibliography

679683

Appendix: Chronological TableIndex

697

A

History of Mathematics

CHAPTER

I

Primitive Origins

Did you bring me a man who cannot numberfingers?

his

From

the

Book of the Dead

Mathematicians of the twentieth century carry on a highly sophisticated which is not easily defined but much of the subject that today is known as mathematics is an outgrowth of thought that originally centered in the concepts of number, magnitude, and form. Old-fashioned definitions of mathematics as a "science of number and magnitude" are no longer valid, but they do suggest the origins of the branches of mathematics. Primitive notions related to the concepts of number, magnitude, and form can be traced back to the earliest days of the human race, and adumbrations of mathematical notions can be found in forms of life that may have antedated mankind by many millions of years. Darwin in Descent of Man (1871) noted that certain of the higher animals possess such abilities as memory and imagination, and today it is even clearer that the abilities to distinguish number, size, order, and form rudiments of a mathematical sense are notintellectual activity;

mankind. Experiments with crows, for example, have shown that at least certain birds can distinguish between sets containing up to four elements. 1 An awareness of differences in patterns found in their environment is clearly present in many lower forms of life, and this is akin to the mathematician's concern for form and relationship. At one time mathematics was thought to be directly concerned with the world of our sense experience, and it was only in the nineteenth century that pure mathematics freed itself from limitations suggested by observations of nature. It is clear that originally mathematics arose as a part of the everyday life of man, and if there is validity in the biological principle of the "survival

exclusively the property of

human race probably is not unrelated development in man of mathematical concepts. At first the primitive notions of number, magnitude, and form may have been related to contraststo theSee Levi Conant, The Number Concept. Its Origin and Development "Animals as Mathematicians," Nature, 202 (1964), 1156-1160.1

of the fittest," the persistence of the

(1923). Cf. H.

Kalmus,

A HISTORY OF MATHEMATICSIndia

Iran

Hoopta*

Syria

Egypt

Asia minor

Greece

Italy

Spain]-S500

1500

Chronological scheme representing the extent of some ancient and medieval civilizations. (Reproduced, with permission, from O. Neugebauer, The Exact Sciences in Antiquity.)

rather than likenesses inequality in size of athe

the

difference between one wolf

and many, themust have

minnow and a whale, the unlikeness of the roundness of

moon and

the straightness of a pine tree. Gradually there

arisen, out of the welter of chaotic experiences, the realization that there are

PRIMITIVE ORIGINS

can be put into one-to-one correspondence. The hands can be matched against the feet, the eyes, the ears, or the nostrils. This recognition of an abstract property that certain groups hold in common, and which we call number, represents a long step toward modern mathematics. It is unlikely to have been the discovery of any one individual or of anysingle tribe;it

this awareness of similarities in number and form both and mathematics were born. The differences themselves seem to point to likenesses, for the contrast between one wolf and many, between one sheep and a herd, between one tree and a forest, suggests that one wolf, one sheep, and one tree have something in common their uniqueness. In the same way it would be noticed that certain other groups, such as pairs,;

samenesses and from

science

was more

probably a gradual awareness which may have developed as early in man's cultural development as his use of fire, possibly some 300,000 years ago That the development of the number concept was a long and gradual process is suggested by the fact that some languages, including Greek, have preserved in their grammar a tripartite distinction between one and two and more than two, whereas most languages today make only the dual distinction in "number" between singular and plural. Evidently our very early ancestors at first counted only to two, any set beyond this level beingstigmatized as

"many." Even today many primitive peoples them into bundles of two each.

still

count objects by arranging

of number ultimately became sufficiently extended and need was felt to express the property in some way, presumably at first in sign language only. The fingers on a hand can be readily used to indicate a set of two or three or four or five objects, the number one generally not being recognized at first as a true "number." By the use of the fingers on both hands, collections containing up to ten elements could be represented; by combining fingers and toes, one could mount as highvivid so that a

The awareness

as twenty.

were inadequate, heaps of stones could be used to represent a correspondence with the elements of another set. Where primitive man used such a scheme of representation, he often piled the stones in groups of five, for he had become familiar with quintuples through observation of the human hand and foot. As Aristotle had noted long ago, the widespread use today of the decimal system is but the result of the anatomical accident that most of us are born with ten fingers and ten toes. From the mathematical point of view it is somewhat inconvenient that Cro-Magnon man and his descendants did not have either four or six fingers on a hand. Although historically finger counting, or the practice of counting by fives and tens, seems to have come later than countercasting by twos and threes, the quinary and decimal systems almost invariably displaced the binary anddigits

the

human

When

ternary schemes.

A

study of several hundred tribes

among

the

American

;

4

A HISTORY OF MATHEMATICS

showed that almost one third used a decimal base and about another third had adopted a quinary or a quinary-decimal system fewer than a third had a binary scheme, and those using a ternary system constituted less than 1 per cent of the group. The vigesimal system, withIndians, for example,

twenty as a base, occurred in about 10 per cent of the tribes. Groups of stones are too ephemeral for preservation of information; hence prehistoric man sometimes made a number record by cutting notches in a stick or a piece of bone. Few of these records remain today, but in Czechoslovakia a bone from a young wolf was found which is deeply incised withfifty-fivefirst

2

notches. These are arranged inthirty in thefive.;

two

series,

and

second within each

series the

with twenty-five in the notches are arranged in

groups ofidea of

Such archaeological discoveries provide evidence that theis

metals or of wheeled vehicles.

than such technological advances as the use of It antedates civilization and writing, in the usual sense of the word, for artifacts with numerical significance, such as the bone described above, have survived from a period of some 30,000 years ago. Additional evidence concerning man's early ideas on number can be found

number

far older

our language today. It appears that our words "eleven" and "twelve" originally meant "one over" and "two over," indicating the early dominance of the decimal concept. However, it has been suggested that perhaps the Indo-Germanic word for eight was derived from a dual form for four, andin

that the Latinit

novem

for nine

may

be related to novus (new) in the sense that

was the beginning of a new sequence. Possibly such words can be interpreted as suggesting the persistence for some time of a quaternary or anscale, just as the French quatre-vingt of today appears to be a remnant of a vigesimal system.

octonary

from other animals most strikingly in his language, the development of which was essential to the rise of abstract mathematical thinking yet words expressing numerical ideas were slow in arising. Number signs probably preceded number words, for it is easier to cut notches in a stick than it is to establish a well-modulated phrase to identify a number. Had the problem of language not been so difficult, rivals to the decimal system might have made greater headway. The base five, for example, was one of the earliest to leave behind some tangible written evidence but by the time that language became formalized, ten had gained the upper hand. The modern languages of today are built almost without exception around thediffers; ;

Man

base ten, so that the

number

thirteen, for example,ten.

is

not described as three

and2

five

and

five,

but as three andof

The

tardiness in the development of

W.

C. Eels,

"Number SystemsCf. also

Monthly, 20 (1913), 293.

D.

J.

Struik, "Stone

North American Indians," American Mathematical Age Mathematics," Scientific American, 179

(December

1948), 44-49.

5

PRIMITIVE ORIGINS

language to cover abstractions such as number is seen also in the fact that primitive numerical verbal expressions invariably refer to specific concrete collections such as "two fishes" or "two clubs" and later some such phrase would be adopted conventionally to indicate all sets of two objects.

The tendencyseen in

language to develop from the concrete to the abstract is The height of a horse is measured in "hands," and the words "foot" and "ell" (or elbow) have similarly been derived from parts of the body.for

many

of our present-day measures of length.

The thousands

of years required for

man

to separate out the abstracttestify to

concepts from repeated concrete situations

the difficulties that

must have been, experienced in laying even a very primitive basis for mathematics. Moreover, there are a great many unanswered questions relating to the origins of mathematics. It usually is assumed that the subject arose in answer to man's practical needs, but anthropological studies suggest the 3 possibility of an alternative origin. It has been suggested that the art of counting arose in connection with primitive religious ritual and that theordinal aspect preceded the quantitative concept. In ceremonial rites depicting creation

myths

it

was necessary to

call the participants

onto the scene

in

a specific order, and perhaps counting

was invented

to take care of this

problem.

If theories

of the ritual origin of counting are correct, the concept

that of the cardinal number. Moreover, such an origin would tend to point to the possibility that counting stemmed from a unique origin, spreading subsequently to other portions of the earth. This view, although far from established, would be in harmony with the ritual division of the integers into odd and even, the former being regarded as male, the latter as female. Such distinctions were known to civilizations in all corners of the earth, and myths regarding the male and female numbers have been remarkably persistent. The concept of whole number is one of the oldest in mathematics, and its origin is shrouded in the mists of prehistoric antiquity. The notion of a rational fraction, however, developed relatively late and was not in general closely related to man's systems for the integers. Among primitive tribes there seems to have been virtually no need for fractions. For quantitative needs the practical man can choose units that are sufficiently small to obviate the necessity of using fractions. Hence there was no orderly advance from binary to quinary to decimal fractions, and decimals were essentially the product of the modern age in mathematics, rather than of the ancient period.

of the ordinal

number may have preceded

Statements about the origins of mathematics, whether of arithmetic or geometry, are of necessity hazardous, for the beginnings of the subject are3

4

See A. Seidenberg, "The Ritual Origin of Counting," Archive for History of Exact Sciences, 2

(1962), 1-40.

6

A HISTORY OF MATHEMATICSIt is

older than the art of writing.in a career that

only during the

last

half-dozen millennia,

may have spanned thousands

of millennia, that

man

has been

and thoughts in written form. For data about the prehistoric age we must depend on interpretations based on the few surviving conjectural artifacts, on evidence provided by current anthropology, and on a Herodotus and Aristotle backward extrapolation from surviving documents.able to put his records

were unwilling to hazard placing origins earlier than the Egyptian civilizaroots of greater tion, but it is clear that the geometry they had in mind had originated in Egypt, for he antiquity. Herodotus held that geometry had believed that the subject had arisen there from the practical need for resurveying after the annual flooding of the river valley. Aristotle argued that it

was the existence of apursuit of geometry.as representing

priestly leisure class in

Egypt that had prompted the

We can look upon the views of Herodotus and Aristotle

two opposing theories of the beginnings of mathematics, one

priestly holding to an origin in practical necessity, the other to an origin in sometimes were leisure and ritual. The fact that the Egyptian geometers support of referred to as "rope-stretchers" (or surveyors) can be used in

either theory, for the ropes

undoubtedly were used both in laying out temples and in realigning the obliterated boundaries. We cannot confidently contramathematics, dict either Herodotus or Aristotle on the motive leading tobutit is

clear that both

men

man mayfor

have had little and designs suggest a concerngeometry. Pottery,

underestimated the age of the subject. Neolithic leisure and little need for surveying, yet his drawingsfor spatial relationships that

paved the way

weaving, and basketry show

instances of congruence

and symmetry, which aresimple sequences

in essence parts of elementary geometry. Moreover, in design, such as that in Fig. 1.1, suggest a sort of applied

FIG. 1.1

group theory, as well as propositions in geometry and arithmetic. The design makes it immediately obvious that the areas of triangles are to each other of consecutive odd as squares on a side, or, through counting, that the sums numbers, beginning from unity, are perfect squares. For the prehistoric period there are no documents, hence it is impossible to trace the evolution

7

PRIMITIVE ORIGINS

like

of mathematics from a specific design to a familiar theorem. But ideas are hardy spores, and sometimes the presumed origin of a concept may be

only the reappearance of a

much more

ancient idea that had lain dormant.

and relationships may and the enjoyment of beauty of form, motives that often actuate the mathematician of today. We would like to think that at least some of the early geometers pursued their work for thefor spatial designsfeeling

The concern of prehistoric man have stemmed from his aesthetic

sheer joy of doing mathematics, rather than as a practical aid in mensuration but there are other alternatives. One of these is that geometry, like counting,

had an origin in primitive ritualistic practice. The earliest geometrical results found in India constituted what were called the Sulvasutras, or "rules of the cord." These were simple relationships that apparently were applied in the construction of altars and temples. It is commonly thought that the geometrical motivation of the "rope-stretchers" in Egypt was more practical than that of their counterparts in India but it has been suggested 4 that both Indian and Egyptian geometry may derive from a common source a proto;

geometry that is related to primitive rites in somewhat the same way in which science developed from mythology and philosophy from theology. We must bear in mind that the theory of the origin of geometry in a secularization of ritualistic practice is by no means established. The development of geometry may just as well have been stimulated by the practical needs of construction and surveying or by an aesthetic feeling for design and order. We can make conjectures about what led men of the Stone Age to count, to measure, and to draw. That the beginnings of mathematics are older than the oldest civilizations is clear. To go further and categorically identify a specific origin in space or time, however, is to mistake conjecture for history. It is best to suspend judgment on this matter and to move on to the safer ground of the history of mathematics as found in the written documents that have come

down

to us.

BIBLIOGRAPHYConant, Levi, The Number Concept.1923). Eels,Its

Origin and Development

(New York Macmillan,:

W. G, "Number SystemsMonthly, 20 (1913), 293.

of North American Indians," American Mathematical

Kalmus, H., "Animals as Mathematicians," Nature, 202 (1964), 1156-1160. Menninger, Karl, Zahlwort und Ziffer: Eine Kulturgeschichte der Zahlen, 2nd(Gottingen Vandenhoeck:

ed.

&

Ruprecht, 1957-1958, 2

vols.).

A. Seidenberg,(1962), 488-527.

"The Ritual Origin of Geometry," Archive for History of Exact

Sciences, 1

8

A HISTORY OF MATHEMATICS

Exact Sciences, Seidenberg, A., "The Ritual Origin of Geometry," Archive for History of1 (1962),

488-527.

Seidenberg, A.,

"The Ritual Origin of Counting," Archive for History of Exact

Sciences,

2 (1962), 1-40.Smeltzer, Donald,

Man and Number (New York Emerson: :

Books,

1958).;

Smith, D.

E.,

paperback History of Mathematics (Boston Ginn, 1923-1925, 2 vols.:

ed.,

New YorkSmith, D.Struik, D.E.,

Dover,

1958).

and Jekuthiel Ginsburg, Numbers and Numerals (Washington, D.C.:

National Council of Teachers of Mathematics, 1958). American, 179 (December 1948), J., "Stone Age Mathematics," Scientific44-^19.

EXERCISES1.

mathematics Describe the type of evidence on which an account of prehistoricciting

is

based,

some

specific instances.if

2.

What

evidence,

any,

is

there that mathematics began with the advent of

man? Do you

think that mathematics antedates3.

man?

List evidences

4.

What

5.

6.7. 8.

and Why? If you had to choose a number base, which would it be? Which do you think came first, number names or number symbols? Why? Why are there few traces of scales from six to nine? What do you think were the first plane and solid geometric figures to be consciously andsystematically studied?

from language for the use at some time of bases other than ten. three, four, five, ten, twenty, are the advantages and disadvantages of the bases two, of a base? sixty? Do you think that these influenced early man in his choice

Why?influential in the rise of early geometry,

9.

Which do you think was moreastronomy or a need

an interest

in

for surveying? Explain.

10.

Which

of the following time divisions was prehistoric

man

likely to notice

the year, the:

month, the week, the day, the hour? Explain.

CHAPTER

II

EgyptSesostris

.

.

.

made a division.

of the

soil

of Egypt

among

the inhabitants

.

.

If the river carried

of a man's

lot, ...

the king sent

away any portion persons to examine, and

determine by measurement the exact extent of theloss. .

to be

From this practice, I think, geometry first came known in Egypt, whence it passed into Greece..

Herodotus

customary to divide the past of mankind into eras and periods, with and characteristics. Such divisions are helpful, although we should always bear in mind that they are only a framework arbitrarily superimposed for our convenience and that the separations in time they suggest are not unbridged gulfs. The Stone Age, a long period preceding the use of metals, did not come to an abrupt end. In fact, the type of culture that it represented terminated much later in Europe than in certain parts of Asia and Africa. The rise of civilizations characterized by the use of metals took place at first in river valleys, such as those in Egypt, Mesopotamia, India, and China hence we shall refer to the earlier portion of the historical period as the "potamic stage." Chronological records of the civilizations in the valleys of the Indus and Yangtze rivers are quite unreliable, but fairly dependable information is available about the peoples living along the Nile and in the "fertile crescent" of the Tigris and Euphrates rivers. Before the end of the fourth millennium B.C. a primitive form of writing was in use in both the Mesopotamian and Nile valleys. There the early pictographic records, through a steady conventionalizing process, evolved into a linearIt is

particular reference to cultural levels

;

order of simpler symbols. In Mesopotamia, where clay was abundant, wedge-shaped marks were impressed with a stylus upon soft tablets which then were baked hard in ovens or by the heat of the sun. This type of writing is known as cuneiform (from the Latin word cuneus or wedge) because of the

shape of the individual impressions. The meaning to be transmitted in cuneiform was determined by the patterns or arrangements of the wedge-shaped impressions. Cuneiform documents had a high degree of permanence hence many thousands of such tablets have survived from antiquity, many of them;

10

A HISTORY OF MATHEMATICS

IA.II!

!

ML7&*=>2^7-&- q, two regular sexagesimal integers, which we shall call p and q, 2 2 - q 2 and 2pq and p + q 2 The and then formed the triple of numbers p triple in three integers thus obtained are easily seen to form a Pythagorean.

which the square of the

largest

is

equal to the

sum

of the squares of the other

39

MESOPOTAMIA

Hence these numbers can be used as the dimensions of the right triangle ABC, with a = p 2 - q 2 and b = 2pq and c = p 3 +

b

>

c,

the three equations

ab

bc

aa'

ab

bc

ab'

ab

bc

ac

define b respectively as the arithmetic, the geometric,*18. All polyhedral

faces

and n

is

and the harmonic mean of a and c. 2 3 numbers are of the form P = an + bn + en, where m is the number of the order. Use this fact to find a and b and c for tetrahedral numbers (m = 4)

and

verify geometrically for n

=

4.

*19. Polyhedral

numbers are found by adding successive polygonal numbers of the same kind. Show how to generalize this procedure to define polytopal numbers in n-dimensional space and find three nontrivial polytopal numbers.

CHAPTER V

The Heroic AgeI

would rather discover one cause than gain the kingof Persia.

dom

Democritus

Accounts of the origins of Greek mathematics center on the so-called Ionian and Pythagorean schools and the chief representative of each Thales and Pythagoras although reconstructions of their thought rest on fragmentary reports and traditions built up during later centuries. To a certain extent

1

this situation prevails

throughout thescientific

fifth

century

B.C.

no extant mathematical or

documents

until the

There are virtually days of Plato in

the fourth century b.c. Nevertheless, during the last half of the fifth century there circulated persistent and consistent reports concerning a handful of

mathematicians who evidently were intensely concerned with problems that formed the basis for most of the later developments in geometry. We shall

Age of Mathematics," for seldom have men with so little to work with tackled mathematical problems of such fundamental significance. No longer was mathematical activity centered almost entirely in two regions nearly at opposite ends of the Greek world it flourished all about the Mediterranean. In what is now southern Italy there were Archytas of Tarentum (born ca. 428 B.C.) and Hippasus of Metapontum (fl. ca. 400 B.C.); at Abdera in Thrace we find Democritus (born ca. 460 B.C.) nearer the center of the Greek world, oneither before or since; ;

therefore refer to this period as the "Heroic

Ellis (born ca. 460 B.C.); and at nearby Athens there lived at various times during the critical last half of the fifth century B.C. three scholars from other regions Hippocrates of Chios (fl. ca. 430 B.C.), Anaxagoras of Clazomenae (t428 b.c), and Zeno of Elea (fl. ca. 450 B.C.). Through the work of these seven men we shall describe the fundamental changes in mathematics that took place a little before the year 400 b.c.:

the Attic peninsula, there

was Hippias of

b.c. was a crucial period in the history of Western opened with the defeat of the Persian invaders and closed with the surrender of Athens to Sparta. Between these two events lay thefifth

The

centuryit

2

civilization, for

69

;

70great

A HISTORY OF MATHEMATICS

Age of Pericles, with its accomplishments in literature and art. The prosperity and intellectual atmosphere of Athens during the century attracted scholars from all parts of the Greek world, and a synthesis of diverse aspectswas achieved. From Ionia came men, such as Anaxagoras, with a practical turn of mind from southern Italy came others, such as Zeno, with stronger metaphysical inclinations. Democritus of Abdera espoused a materialistic view of the world, while Pythagoras in Italy held idealistic attitudes in science and philosophy. At Athens one found eager devotees of old and new branches of learning, from cosmology to ethics. There was a bold spirit of free inquiry that sometimes came into conflict with established mores. In particular, Anaxagoras was imprisoned at Athens for impiety in asserting that the sun was not a deity, but a huge red-hot stone as big as the whole Peloponnessus, and that the moon was an inhabited earth that borrowed its light from the;

sun.

He

aim of his

well represents the spirit of rational inquiry, for he regarded as the that life the study of the nature of the universea purposefulness

The

he derived from the Ionian tradition of which Thales had been a founder. intellectual enthusiasm of Anaxagoras was shared with his countrymen through the first scientific best-sellera book On Nature which could be

bought in Athens for only a drachma. Anaxagoras was a teacher of Pericles, who saw to it that his mentor ultimately was released from prison. Socrates was at first attracted to the scientific ideas of Anaxagoras, but the gadfly of Athens found the naturalistic Ionian view less satisfying than the search forethical verities.

Greek science had been rooted

in a highly intellectual curiosity

which

immediacy of pre-Hellenic thought often is the desire to clearly represented the typical Greek motive Anaxagoras In mathematics also the Greek attitude differed sharply from that of know. the earlier potamic cultures. The contrast was clear in the contributions generally attributed to Thales and Pythagoras, and it continues to show through in the more reliable reports on what went on in Athens during thecontrasted with the utilitarian

Heroic Age. Anaxagoras was primarily a natural philosopher rather than a mathematician, but his inquiring mind led him to share in the pursuit of mathematical problems. We are told by Plutarch that while Anaxagoras was in prison he occupied himself in an attempt to square the circle. Here we have the first mention of a problem that was to fascinate mathematicians for more 1 than 2000 years. There are no further details concerning the origin of the problem or the rules governing it. At a later date it came to be understood that the required square, exactly equal in area to the1

circle,

was

to be

W. Hobson, Squaring the Circle (ca. 1913), p. 14. This work has been reprinted several The accuracy of Plutarch's statement in this connection has been questioned recently. On the work of Anaxagoras see D. E. Gershenson and D. A. Greenberg, Anaxagoras and the Birth of Physics (New York: Blaisdell, 1964).See E.times.

71

THE HEROIC AGE

constructed by the use of compasses and straightedge alone. Here we see a type is quite unlike that of the Egyptians and Babylonians. It is not the practical application of a science of number to a facet of lifeof mathematics thatexperience, but a theoretical question involving a nice distinction between

accuracy in approximation and exactitude in thought. The mathematical problem that Anaxagoras here considered was no more the concern of the technologist than were those he raised in science concerning the ultimate structure of matter. In the Greek world mathematics was more closely related to philosophy than to practical affairs, and this kinship has persistedto the present day.

Anaxagoras died in 428 B.C., the year that Archytas was born, just one year before Plato's birth and one year after Pericles' death. It is said that Pericles died of the plague that carried off perhaps a quarter of the Athenianpopulation, and the deep impression that this catastrophe created is perhaps the origin of a second famous mathematical problem. It is reported that a delegation had been sent to the oracle of Apollo at Delos to inquire how the plague could be averted, and the oracle had replied that the cubical altar

Apollo must be doubled. The Athenians are said to have dutifully doubled was of no avail in curbing the plague. The altar had, of course, been increased eightfold in volume, rather than twofold. Here, according to the legend, was the origin of the "duplication of the cube" problem, one that henceforth was usually referred to as the "Delian problem" given the edge of a cube, construct with compasses and straightedge alone the edge of a second cube having double the volumeto

the dimensions of the altar, but this

At about the same time there circulated in Athens still a third problem given an arbitrary angle, construct by means of compasses and straightedge alone an angle one-third as large as the given angle. These three problems the squaring of the circle, the duplication of the cube, and the trisection of the angle have since been known as the "three famous (or classical) problems" of antiquity. More than 2200 years later it was to be proved that all three of the problems were unsolvable by means of straightedge and compasses alone. Nevertheless, the better part of Greek mathematics, and of much later mathematical thought, was suggested by efforts to achieve the impossible or, failing this, to modify the rules. The Heroic Age failed in its immediate objective, under the rules, but the efforts were crowned with brilliant success in other respects.of thefirst.

celebrated

the

Somewhat younger than Anaxagoras, and coming originally from about same part of the Greek world, was Hippocrates of Chios. He should not be confused with his still more celebrated contemporary, the physician Hippocrates of Cos. Both Cos and Chios are islands in the Dodecanese;

group but Hippocrates of Chios

in

about 430

B.C. left his native

land for

:

72

A HISTORY OF MATHEMATICS

Athens in his capacity as a merchant. Aristotle reports that Hippocrates was less shrewd than Thales and that he lost his money in Byzantium through fraud others say that he was beset by pirates. In any case, the incident was never regretted by the victim, for he counted this his good fortune in that he achieved as a consequence he turned to the study of geometry, in which Proclus wrote that remarkable successa story typical of the Heroic Age. Hippocrates composed an "Elements of Geometry," anticipating by more than a century the better-known Elements of Euclid. However, the textbook as well as another reported to have been written by Leon, of Hippocrates;

a later associate of the Platonic

schoolhas been lost, although it was known mathematical treatise from the fifth century has to Aristotle. In fact, no survived but we do have a fragment concerning Hippocrates which SimpliMathematics cius (fl. ca. 520) claims to have copied literally from the History of Eudemus. This brief statement, the nearest thing we have to an (now lost) by;

portion of the original source on the mathematics of the time, describes a work of Hippocrates dealing with the quadrature of lunes. A lune is a figure bounded by two circular arcs of unequal radii the problem of the quadrature Eudemian of lunes undoubtedly arose from that of squaring the circle. The;

fragment attributes to Hippocrates the following theoremSimilar segments of circles are in the same ratio as the squares on their bases.

The Eudemian account

reports that Hippocrates demonstrated this by first showing that the areas of two circles are to each other as the squares on and concept of their diameters. Here Hippocrates adopted the language large a role in Pythagorean thought. In fact, it proportion which played so

is

thought by some that Hippocrates became a Pythagorean. The Pythagorean school in Croton had been suppressed (possibly because of its secrecy, perhaps because of its conservative political tendencies), but the scattering the of its adherents throughout the Greek world served only to broaden or influence of the school. This influence undoubtedly was felt, directlyindirectly,

by Hippocrates.of Hippocrates

The theorem

precise believed that Hippocrates gave a proof of the theorem, but a rigorous demonstration at that time (say about 430 B.C.) would appear to be unlikely.

on the areas of circles seems to be the earliest statement on curvilinear mensuration in the Greek world. Eudemus

The theory of proportions

at that stage

mensurable magnitudes Eudoxus, a man who lived halfway between Hippocrates and Euclid. Howto ever, just as much of the material in the first two books of Euclid seems stem from the Pythagoreans, so it would appear reasonable to assume that the formulations, at least, of much of Books III and IV of the Elements came from the work of Hippocrates. Moreover, if Hippocrates did give aonly.

The proof as given

probably was established for comin Euclid XII. 2 comes from

73

THE HEROIC AGE

demonstration of

his theorem on the areas of circles, he may have been responsible for the introduction into mathematics of the indirect method of

proof. That is, the ratio of the areas of two circles is equal to the ratio of the squares on the diameters or it is not. By a reductio ad absurdum from the second of the two possibilities, the proof of the only alternative is established.

Fromfirst

He

his theorem on the areas of circles Hippocrates readily found the rigorous quadrature of a curvilinear area in the history of mathematics. began with a semicircle circumscribed about an isosceles right triangle,

and on the base (hypotenuse) he constructed a segment similar to the circular segments on the sides of the right triangle (Fig. 5.1). Because the segments

are to each other as squares on their bases, and from the Pythagorean theorem as applied to the right triangle, the sum of the two small circular segments is equal to the larger circular segment. Hence the difference between the semicircle on AC and the segment ADCE equals triangle ABC.

ABCD is precisely equal to triangle ABC; and since equal to the square on half of AC, the quadrature of the lune has been found. 2Therefore the lunetriangle

ABC

is

Eudemus

ABCD AD is equal to the sum of the squares on the three equal shorter sides AB and BC and CD (Fig. 5.2). Then if on side AD one constructs a circular segment AEDF similar to those on the three equal sides, lune ABCDE is equal to trapezoid ABCDF.isosceles trapezoid

describes also an Hippocratean lune quadrature based on an inscribed in a circle so that the square on the

longest side (base)

That we are on

relatively firm

ground

historically in describing the

quadrature of lunes by Hippocrates, is indicated by the fact that scholars other than Simplicius also refer to this work. Simplicius lived in the sixth century, but he depended not only on Eudemus (fl. ca. 320 B.C.) but also on Alexander of Aphrodisias (fl. ca. a.d. 200), one of the chief commentators on2

An excellent account of Hippocrates' quadratures is found in B. L. van der Waerden, Science(1961), pp. 131ff.

Awakening

74

A HISTORY OF MATHEMATICS

B

Aristotle.(1)

Alexander describes two quadratures other than those given above. triangle one constructs If on the hypotenuse and sides of an isosceles right5.3),

semicircles (Fig.

then the lunes created on the smaller sides together on a diameter of a semicircle one constructs an three equal isosceles trapezoid with three equal sides (Fig. 5.4), and if on theequal the triangle.(2) If

sides semicircles are constructed, then the trapezoid:

is

equal in area to the

sum of four curvilinear areas the three equal lunes and a semicircle on one quadratures it of the equal sides of the trapezoid. From the second of these hence the if the lunes can be squared, the semicircle would follow that be squared. This conclusion seems to have encouraged circlecan alsoHippocrates, as well as his contemporaries and early successors, to hope that ultimately the circle would be squared.

The Hippocratean quadratures

are significant not so

much

as attempts at

circle-squaring as indications of the level of mathematics at the time.

They

show that Athenian mathematicians were adept at handling transformationsof areas

and proportions. In

particular, there

was evidently no

difficulty in

the converting a rectangle of sides a and b into a square. This required finding or geometric mean between a and b. That is, if a :x = x :b, mean proportional

geometers of the day easily constructed the line x. It was natural, therefore, two means that geometers should seek to generalize the problem by inserting

75

THE HEROIC AGE

between two given magnitudes a and b. That is, given two line segments a and b, they hoped to construct two other segments x and y such that a:x = x:y = y.b. Hippocrates is said to have recognized that this problem is equivalent to that of duplicating the cube for if b = 2a, the continued;

proportions,

the elimination of y, lead to the conclusion that x 3 = 2a 3 There are three views on what Hippocrates deduced from his quadrature

upon

.

of lunes. Some have accused him of believing that he could square all lunes, hence also the circle others think that he knew the limitations of his work, concerned as it was with some types of lunes only. At least one scholar has held that Hippocrates knew he had not squared the circle but tried to deceive;

countrymen into thinking that he had succeeded. 3 There are other questions, too, concerning Hippocrates' contributions, for to him has beenhis

ascribed, withIt is

some

uncertainty, the

first

use of

letters in

geometric figures.

interesting to note that whereas he

advanced two of the three famousEllis.

problems, he seems to havea problem studied

made nolater

progress in the trisection of the angle,

somewhat

by Hippias of

Toward the end of the fifth century B.C. there flourished at Athens a group of professional teachers quite unlike the Pythagoreans. Disciples of Pythagoras had been forbidden to accept payment for sharing their knowledge withothers.

The

Sophists, however, openly supported themselves

honest intellectual endeavor, but also in the art of "making the worse appear the better." To a certain extent the accusation of shallowness directed against the Sophists was warranted ; but this should not conceal the fact that Sophists usually were very widely informed in manyinfields

fellow citizens

not only

by tutoring

these

and that some of them made real contributions to learning. Among was Hippias, a native of Ellis who was active at Athens in the second

half of the fifth century B.C.

He is one

we have

firsthand information, for

we

of the earliest mathematicians of whom learn much about him from Plato's

had made much, from mathematics to oratory, but none of his work has survived. He had a remarkable memory, he boasted immense learning, and he was skilled in handicrafts. To this Hippias (there were many others in Greece who bore the same name) we apparently owe the introduction into mathematics of the first curve beyond the circle and the straight line. Proclus and other commentators ascribe to him the curve since known as the trisectrix or quadratrix

dialogues.

We

read, for example, that Hippias boasted that he

more money than any two other

Sophists.

He

is

said to have written

of Hippias.3

4

This

is

drawn

as follows

:

In the square

ABCD (Fig.

5.5) let side

See Bjornbo's article "Hippocrates" in Pauly-Wissowa, Real-Enzyklopiidie der klassischen

Altertumswissenschaft, Vol. VIII, p. 1796. 4 An excellent account of this is found in Kathleen Freeman, The Pre-Socratic Philosophers. A Companion to Diels, Fragmente der Vorsokratiker (1949), pp. 381-391. See also the article on

Hippias in Pauly-Wissowa,

op.

cit.,

VIII, 1707

ff.

76

A HISTORY OF MATHEMATICS

AB move down DC and let this

uniformly from

its

present position until

it

coincides with

that side DA motion position until it coincides with DC. If rotates clockwise from its present the positions of the two moving lines at any given time are given by A'B' and DA" respectively and if P is the point of intersection of A'B' and DA", curve the locus of P during the motions will be the trisectrix of Hippias in the figure. Given this curve, the trisection of an angle is carried out APQ with ease. For example, if PDC is the angle to be trisected, one simply trisects segments B'C and AD at points R, S, T, and U. If lines TR and US respectively, lines VD and WD will, by the cut the trisectrix in V and

take place in exactly the

same time

W

property of the

trisectrix, divide

angleis

PDC

in three equal parts.

The curve

of Hippias generallycircle.

known

as the quadratrix, since

be used to square thethis application

Whether

or not Hippias himself

it can was aware of

be determined. It has been conjectured that Hippias knew of this method of quadrature but that he was unable to justify specifically given later it. Since the quadrature through Hippias' curve was

cannot

now

by Dinostratus, we shall describe this work in the next chapter. Hippias lived at least as late as Socrates (t399 B.C.), and from the pen of Plato we have an unflattering account of him as a typical Sophist vain, boastful, and acquisitive. Socrates is reported to have described Hippias as

handsome and

learned, but boastful and shallow. Plato's dialogue on Hippias satirizes his show of knowledge, and Xenophon's Memorabilia includes an unflattering account of Hippias as one who regarded himself an

expert in everything from history and literature to handicrafts and science. In judging such accounts, however, we must remember that Plato and

Xenophon were uncompromisingly opposed to the Sophists in general. It is well to bear in mind also that both Protagoras, the "founding fatherof the Sophists," and Socrates, the archopponent of the movement, were

77

THE HEROIC AGE

antagonistic to mathematics and the sciences. With respect to character, Plato contrasts Hippias with Socrates, but one can bring out much the samecontrast by comparing Hippias with another contemporary

the Pythagor-

ean mathematician Archytas of Tarentum.

Pythagoras is said to have retired to Metapontum toward the end of his and to have died there about 500 B.C. Tradition holds that he left no written works, but his ideas were carried on by a large number of eagerlife

The center at Croton was abandoned when a rival political group from Sybaris surprised and murdered many of the leaders, but those who escaped the massacre carried the doctrines of the school to other parts of the Greek world. Among those who received instruction from the refugees was Philolaus of Tarentum, and he is said to have written the first account of Pythagoreanism permission having been granted, so the story goes, todisciples.

fortunes. Apparently it was this book from which Plato derived his knowledge of the Pythagorean order. The number fanaticism that was so characteristic of the brotherhood evidently was shared by Philolaus,

repair his

damaged

was from his account that much of the mystical lore concerning the was derived, as well as knowledge of the Pythagorean cosmology. The Philolaean cosmic scheme is said to have been modified by two later Pythagoreans, Ecphantus and Hicetas, who abandoned the central fire and counterearth and explained day and night by placing a rotating earth at the center of the universe. The extremes of Philolaean number worship also seem to have undergone some modification, more especially at the handsandit

tetractys

of Archytas, a student of Philolaus at Tarentum.sect had exerted a strong intellectual influence throughGraecia, with political overtones that may be described as a sort of "reactionary international," or perhaps better as a cross between Orphism

The Pythagorean

out

Magna

at outlying

and Freemasonry. At Croton political aspects were especially noticeable, but Pythagorean centers, such as Tarentum, the impact was primarily;

intellectual. Archytas believed firmly in the efficacy of number his rule of the city, which allotted him autocratic powers, was just and restrained, for

he regarded reason as a force working toward social amelioration. For many was elected general, and he was never defeated yet he was kind and a lover of children, for whom he is reported to have invented "Archytas' rattle." Possibly also the mechanical dove, which he is said to have fashioned of wood, was built to amuse the young folk. Archytas continued the Pythagorean tradition in placing arithmetic above geometry, but his enthusiasm for number had less of the religious and mystical admixture found earlier in Philolaus. He wrote on the application of theyears in succession he;

arithmetic, geometric,either Philolaus or

and subcontrary means to music, and it was probably Archytas who was responsible for changing the name of

78

A HISTORY OF MATHEMATICS

one to "harmonic mean." Among his statements in this connection was the observation that between two whole numbers in the ratio n:(n + 1) there could be no integer that is a geometric mean. Archytas gave more attention to music than had his predecessors, and he felt that this subjectthe last

should play a greater role than literaturehis conjectures

in the

education of children.

Among

was one that attributed

differences in pitch to varying rates

of motion resulting from the flow causing the sound. Archytas seems to have paid considerable attention to the role of mathematics in the curriculum, and to him has been ascribed the designation of the four branches in the mathematical quadrivium arithmetic (or numbers at rest), geometry (or magnitudes at rest), music (or numbers in motion), and astronomy (or magnitudes in motion). These subjects, together with the trivium consisting of grammar, rhetoric, and dialectics (which Aristotle traced back to Zeno), role that later constituted the seven liberal arts; hence the prominent

mathematics has played

in

education

is

in

no small measure due

to Archytas.

8

Archytas had access to an earlier treatise on the elements of mathematics, and the iterative square-root process often known by the name of Archytas had been used long before in Mesopotamia. Nevertheless, Archytas was himself a contributor of original mathematical results. The most striking contribution was a three-dimensional solution of the DelianIt is

likely that

problem which may be mostin the

easily described, somewhat anachronistical^, of analytic geometry. Let a be the edge of the cube three mutually to be doubled, and let the point (a, 0, 0) be the center of plane perpendicular perpendicular circles of radius a and each lying in a

modern language

to a coordinate axis.

the circle perpendicular to the x-axis construct xy-plane a right circular cone with vertex (0, 0, 0); through the circle in the circle in the xz-plane be revolved pass a right circular cylinder and let the about the z-axis to generate a torus. The equations of these three surfaces

Through

;

2 2 2 2 2 = x 2 = y 2 + z and lax = x + y and (x + y + z ) 2 2 2 three surfaces intersect in a point whose x-coordinate is 4a (x + y ). These of the cube desired. crfl; hence the length of this line segment is the edge 2 2

are respectively

his solution

The achievement of Archytas is the more impressive when we recall that was worked out synthetically without the aid of coordinates.

Nevertheless, the most important contribution of Archytas to mathematics may have been his intervention with the tyrant Dionysius to save the life of his friend, Plato. The latter remained to the end of his life deeply committed to the Pythagorean veneration of number and geometry, and the

supremacy of Athens in the mathematical world of the fourth century B.C. resulted primarily from the enthusiasm of Plato, the "maker of mathematicians." However, before taking up the role of Plato it is necessary to discuss the work of an earlier Pythagoreanan apostate by the name of Hippasus.

79

THE HEROIC AGE

Hippasus of Metapontum (or Croton), roughly contemporaneous with is reported to have been originally a Pythagorean but to have been expelled from the brotherhood. One account has it that the Pythagoreans erected a tombstone to him, as though he were dead; another story reports that his apostasy was punished by death at sea in a shipwreck. ThePhilolaus,

exact cause of the break

is unknown, in part because of the rule of secrecy, but there are three suggested possibilities. According to one, Hippasus was expelled for political insubordination, having headed a democratic move-

rule. A second tradition attributes the expulsion to disclosures concerning the geometry of the pentagon or the dodecahedron perhaps a construction of one of the figures. A third

ment against the conservative Pythagorean

explanation holds that the expulsion was coupled with the disclosure of a mathematical discovery of devastating significance for Pythagorean philo-

sophyItall

the existence of incommensurable magnitudes.

things, in

had been a fundamental tenet of Pythagoreanism that the essence of geometry as well as in the practical and theoretical affairs of

man, are explainable in terms of arithmos, or intrinsic properties of whole numbers or their ratios. The dialogues of Plato show, however, that the Greek mathematical community had been stunned by a disclosure thatvirtually

demolished the basis

for the

Pythagorean

faith in

whole numbers.

This was the discovery that within geometry itself the whole numbers and their ratios are inadequate to account for even simple fundamental properties. They do not suffice, for example, to compare the diagonal of a square or a

cube or a pentagon with its side. The line segments are incommensurable, no matter how small a unit of measure is chosen. Just when and how the discovery was made is not known, but much ink has been spilled in support of one hypothesis or another. Earlier arguments in favor of a Hindu origin of the discovery 5 lack foundation, and there seems to be little chance that Pythagoras himself was aware of the problem of incommensurability. The most plausible suggestion is that the discovery was made by the later Pythagoreans at some time before 410 B.C. 6 Some would attribute it specifically to Hippasus of Metapontum during the earlier portion of the last quarter ofthe5

fifth

century

B.C.,

7

while others place

it

about another half a century

later.

See Heinrich Vogt,

"Haben

die alten Inder den Pythagoreischen Lehrsatz(3),1

tionale gekannt?,"6

Bibliotheca Mathematica

und das Irra7 (1906-1907), 6-23; also Leopold von

Schroeder, Pythagoras und die Inder (Leipzig.

884).

See especially Heinrich Vogt, "Die Entdeckungsgeschichte des Irrationalen nach Plato und anderen Quellen des 4. Jahrhunderts," Bibliotheca Mathematica (3), 10 (1910), 97-155, and the same author's paper, "Zur Entdeckungsgeschichte des Irrationalen," Bibliotheca Mathematica (3), 14 (1914), 9-29. Cf. Heath, History of Greek Mathematics (1921), 1, 157. 7 See Kurt von Fritz, "The Discovery of Incommensurability by Hippasus of Metapontum," Annals of Mathematics (2), 46 (1945), 242-264.

80

A HISTORY OF MATHEMATICSThe circumstances surroundingthe earliest recognition of incommensuris

able line segments are as uncertain asit is

the time of the discovery. Ordinarilyin

assumed that the of the Pythagorean theorem

recognition came

connection with the application

to the isosceles right triangle. Aristotle refers

to a proof of the incommensurability of the diagonal of a square with respectit was based on the distinction between odd and Such a proof is easy to construct. Let d and s be the diagonal and side that is, that the of a square, and assume that they are commensurable radio d/s is rational and equal to p/q, where p and q are integers with no common factor. Now, from the Pythagorean theorem it is known that 2 2 2 2 2 2 2 2 hence (d/s) 2 = p /q = 2, or p = 2q Therefore p must be d = s + s even hence p must be even. Consequently q must be odd. Letting p = 2r 2 2 2 2 2 2 and substituting in the equation p = 2q we have 4r = 2q or q = 2r 2 even. However, q was shown above Then q must be even hence q must be to be odd, and an integer cannot be both odd and even. It follows therefore, by the indirect method, that the assumption that d and s are commensurable must be false.

to a side, indicating that

even.

8

;

.

;

,

,

.

;

10it

In this proof the degree of abstraction

is

so high that the possibility that

was the

basis for the original discovery of incommensurability has been

questioned. There are, however, other ways in which the discovery could have come about. Among these is the simple observation that when the five

diagonals of a regular pentagon are drawn, these diagonals form a smaller regular pentagon (Fig. 5.6), and the diagonals of the second pentagon in

FIG. 5.6

turn form a third regular pentagon, which

is still

smaller. This process can

be continued indefinitely, resulting in pentagons that are as small as desired and leading to the conclusion that the ratio of a diagonal to a side in a regular pentagon is not rational. The irrationality of this ratio is, in fact, a consequence8 See H. G. Zeuthen, "Sur l'origine historique de la connaissance des quantites irrationelles," Oversigt over del Kongelige Danske Videnskabernes Selskabs. Forhandlinger, 1915, pp. 333-362.

81

THE HEROIC AGE

argument presented in connection with Fig. 4.2 in which the golden was shown to repeat itself over and over again. Was it perhaps this property that led to the disclosure, possibly by Hippasus, of incommensurability? There is no surviving document to resolve the question, but the suggestion is at least a plausible one. In this case, it would not have been y/2 but ^/5 that first disclosed the existence of incommensurable magnitudes, for the solution of the equation a:x = x:{a x) leads to (^/S l)/2 as the ratio of the side of a regular pentagon to a diagonal. The ratio of the diagonal of a cube to an edge is y/3, and here, too, the spectre of the incommensurableof thesectionrearsits

ugly head.side can be provided also for the ratio of the diagonal of a

A geometric proof somewhat analogous to that for the ratio of the diagonalof a pentagon toits

square to

its side. If in

the square

ABCD (Fig. 5.7) one lays off on the diagonal

FIG. 5.7

AC the segment AP = AB and at P erects the perpendicular PQ, the ratio CQ to PC will be the same as the ratio of A C to AB. Again, if on CQ one lays off QR = QP and constructs RS perpendicular to CR, the ratio ofof

hypotenuse to side again will be what it was before. This process, too, can be continued indefinitely, thus affording a proof that no unit of length,

however small, can be found so that the hypotenuse and amensurable.

side will be

com-

The Pythagorean doctrine that "Numbers constitute the entire heaven" was now faced with a very serious problem indeed but it was not the only one, for the school was confronted also by arguments propounded by the neighboring Eleatics, a rival philosophical movement. Ionian philosophers of Asia Minor had sought to identify a first principle for all things. Thales had thought to find this in water, but others preferred to think of air or fire;

11

82

A HISTORY OF MATHEMATICS

as the basic element.

postulating that

The Pythagoreans had taken a more abstract direction, number in all its plurality was the basic stuff behind phenom-

ena; this numerical atomism, beautifully illustrated in the geometry of figurate numbers, had come under attack by the followers of Parmenides(fi. ca. 450 B.C.). The fundamental tenet of the Eleatics was the unity and permanence of being, a view that contrasted with the Pythagorean ideas of multiplicity and change. Of Parmenides' disciples the best known was Zeno the Eleatic (fi. ca. 450 B.C.) who propounded arguments to prove the inconsistency in the concepts of multiplicity and divisibility. The method Zeno adopted was dialectical, anticipating Socrates in this indirect mode of argument: starting from his opponent's premises, he reduced these to an

of Elea

absurdity.

The Pythagoreans had assumed that space and time can be thought of and instants but space and time have also a property, more easily intuited than defined, known as "continuity." The ultimate elements making up a plurality were assumed on the one hand to have theas consisting of points;

characteristics of the geometrical unit

the pointand on the other to have

certain characteristics of the numerical units or numbers. Aristotle described a Pythagorean point as "unity having position" or as "unity considered in

that it was against such a view that Zeno propounded his paradoxes, of which those on motion are cited most frequently. As they have come down to us, through Aristotle and others, four of them seem to have caused the most trouble: (1) the Dichotomy, (2) the Achilles, (3) the Arrow, and (4) the Stade. The first argues that before a moving object can travel a given distance, it must first travel half this distance but before it can cover this, it must travel the first quarter of the distance and before this, the first eighth, and so on through an infinite number of subdivisions. The runner wishing to get started, must make an infinite number of

space."

It

has been suggested

9

;

;

it is impossible to exhaust an infinite collection, beginning of motion is impossible. The second of the paradoxes hence the is similar to the first except that the infinite subdivision is progressive rather than regressive. Here Achilles is racing against a tortoise that has been given a headstart, and it is argued that Achilles, no matter how swiftly he may run, can never overtake the tortoise, no matter how slow it may be. By the time

contacts in a finite time but;

have reached the initial position of the tortoise, the latter have advanced some short distance; and by the time that Achilles will have covered this distance, the tortoise will have advanced somewhatthat Achilles willwill

farther;

and so the process continues

indefinitely,

with the result that the

swift Achilles can never overtake the slow tortoise.9 See Paul Tannery, La geometrie grecque (1887), pp. 217-261. For a different view see B. L. van der Waerden, "Zenon und die Grundlagenkrise der griechischen Mathematik," Mathe-

matische Annalen, 117 (1940), 141-161.

:

:

83

THE HEROIC AGE

The Dichotomy and the Achilles argue that motion is impossible under the assumption of the infinite subdivisibility of space and time the Arrow and the Stade, on the other hand, argue that motion is equally impossible if one makes the opposite assumption that the subdivisibility of space and;

time terminates in indivisibles. In the Arrow Zeno argues that an object in flight always occupies a space equal to itself; but that which always occupies a space equal to itself is not in motion. Hence the flying arrow is at rest atall

times, so that

its

motion

is

an

illusion.

Mostdescribe,

controversial of the paradoxesis

the Stade (or Stadium), but thex, , ,

on motion, and most awkward to argument can be phrased somewhat

A A 2 A 3 A 4 be bodies of equal size that are stationary; B 2 B 3 BA be bodies, of the same size as the ^4's, that are moving to the right so that each B passes each A in an instant the smallest possible interval of time. Let C C 2 C 3 C 4 also be of equal size with the A's andas follows. Letlet

B^

,

,

,

t

,

,

,

B's

eachthe

them move uniformly to the left with respect to the A's so that in an instant of time. Let us assume that at a given time bodies occupy the following relative positionsandlet

C passes each A

A

x

A2

A,

AA

Bx

B2

B3

B4

Cx

C2

c3is,

C4after

Then

after the lapse of

division of time

the positionsAx

a single instantwill

that

an

indivisible sub-

be as follows

A2

A3

A*

Bx

B2

B3

BA

Cx

C2

c3

C4

;

84

A HISTORY OF MATHEMATICS

then, that C t will have passed two of the B's; hence the instant cannot be the minimum time interval, for we can take as a new and smaller unit the time it takes C l to pass one of the B's. 10 The arguments of Zeno seem to have had a profound influence on the development of Greek mathematics, comparable to that of the discovery of the incommensurable, with whic